C ++中的数值积分

Question

我需要集成一个函数（两个变量）。我知道我可以通过使用Fubini定理来集成一个变量函数，然后使用数值方法，如Rectangle方法或梯形规则。

但是在C ++中有没有预先构建的函数呢？我需要整合单位R2三角形((0,0), (1,0), (0,1))。

Answer 1

您可以使用GNU Scientific Library，它支持许多“数值分析”功能，包括集成。

手册中非常简单的example of integration只是几行代码：

#include <stdio.h>
#include <math.h>
#include <gsl/gsl_integration.h>

double f (double x, void * params) {
  double alpha = *(double *) params;
  return log(alpha*x) / sqrt(x);
}

int
main (void)
{
  double result, error;
  double expected = -4.0;
  double alpha = 1.0;
  gsl_integration_workspace * w 
    = gsl_integration_workspace_alloc (1000);

  gsl_function F;
  F.function = &f;
  F.params = &alpha;

  gsl_integration_qags (&F, 0, 1, 0, 1e-7, 1000,
                        w, &result, &error); 

  printf ("result          = % .18f\n", result);
  printf ("exact result    = % .18f\n", expected);
  printf ("estimated error = % .18f\n", error);
  printf ("actual error    = % .18f\n", result - expected);
  printf ("intervals =  %d\n", w->size);

  gsl_integration_workspace_free (w);

  return 0;
}

Answer 2

据我所知，在标准库中没有您要搜索的类型的函数。这是一个实现：

扩展的梯形规则。

对于固定函数f(x)要在固定极限a和b之间进行积分，可以使扩展梯形规则中的间隔数加倍，而不会失去先前工作的好处。梯形规则的最粗糙实现是在其端点a和b处平均函数。细化的第一阶段是在中间点将函数的值加到该平均值。第二阶段的改进是在1/4和3/4点添加值。

许多基本正交算法涉及添加连续的细化阶段。将此特征封装在Quadrature结构中很方便：

struct Quadrature
{  
   //Abstract base class for elementary quadrature algorithms.
   Int n; // Current level of refinement.

   virtual Doub next() = 0;
   //Returns the value of the integral at the nth stage of refinement. 
   //The function next() must be defined in the derived class.
};

然后Trapzd结构由此得出如下：

template<class T> 
struct Trapzd: Quadrature
{
    Doub a, b, s; // Limits of integration and current value of integral.
    T &func;

    Trapzd() { };

    // func is function or functor to be integrated between limits: a and b 
    Trapzd(T &funcc, const Doub aa, const Doub bb)   
        : func(funcc), a(aa), b(bb)
    {
        n = 0;
    }

    // Returns the nth stage of refinement of the extended trapezoidal rule. 
    // On the first call (n = 1), the routine returns the crudest estimate  
    // of integral of f x / dx in [a,b]. Subsequent calls set n=2,3,... and
    // improve the accuracy by adding 2n - 2 additional interior points.
    Doub next()
    {
        Doub x, tnm, sum, del;
        Int it, j;
        n++;

        if (n == 1)
        {
            return (s = 0.5 * (b-a) * (func(a) + func(b)));
        } 
        else
        {
            for (it = 1, j = 1; j < n - 1; j++)
            {
                it <<= 1;
            }
            tnm = it;
            // This is the spacing of the points to be added.          
            del = (b - a) / tnm; 
            x = a + 0.5 * del;

            for (sum = 0.0,j = 0; j < it; j++, x += del)
            {
                sum += func(x);
            }
            // This replaces s by its refined value.  
            s = 0.5 * (s + (b - a) * sum / tnm); 
            return s;
        }
    }
};

用法：

Trapzd结构可以以多种方式使用。最简单和最粗糙的是通过扩展的梯形规则集成函数，您可以事先知道所需的步数。如果你想要2^M + 1，你可以通过片段完成这个：

Ftor func;      // Functor func here has no parameters.
Trapzd<Ftor> s(func, a, b);
for(j = 1 ;j <= m + 1; j++) val = s.next();

回答为val答案。这里Ftor是一个包含要集成功能的仿函数。

奖金：

当然，更好的是改进梯形法则，直到达到一定程度的准确度。这个功能是：

template<class T>
Doub qtrap(T &func, const Doub a, const Doub b, const Doub eps = 1.0e-10)
{
    // Returns the integral of the function or functor func from a to b. 
    // The constants EPS can be set to the desired fractional accuracy and    
    // JMAX so that 2 to the power JMAX-1 is the maximum allowed number of   
    // steps. Integration is performed by the trapezoidal rule.

    const Int JMAX = 20;
    Doub s, olds = 0.0; // Initial value of olds is arbitrary.

    Trapzd<T> t(func, a, b);

    for (Int j = 0; j < JMAX; j++) 
    {
        s = t.next();

        if (j > 5) // Avoid spurious early convergence.
        {
            if (abs(s - olds) < eps * abs(olds) || (s == 0.0 && olds == 0.0)) 
            {
                return s;
            }
        }
        olds = s;
    }
    throw("Too many steps in routine qtrap");
}

类型定义。

typedef double Doub;    // 64 - bit floating point
typedef int Int;        // 32 - bit signed integer

Answer 3

我推荐高斯求积和线性形状函数：

http://www.mathworks.com/matlabcentral/fileexchange/9230-gaussian-quadrature-for-triangles

http://www.wolframalpha.com/input/?i=gaussian+quadrature+triangle

http://www.cs.rpi.edu/~flaherje/pdf/fea6.pdf

Answer 4

您可能希望查看Boost积分和微分库。特别是，它们提供了梯形规则的一个版本：

https://www.boost.org/doc/libs/1_69_0/libs/math/doc/html/math_toolkit/trapezoidal.html

正交/微分库编写得非常好，并且与现代C ++兼容，因为可以使用lambda表达式或函数对象来实现被积函数。我很快就开始使用它了。

这是一个近似pi的例子，通过近似4 /（1 + x ^ 2）的积分，从x = 0到x = 1，将被积函数作为lambda表达式。

#include <boost/math/quadrature/trapezoidal.hpp>
#include <iostream>
using boost::math::quadrature::trapezoidal;
using std::cout;
using std::endl;

// Put inside a test function:
auto f = [](double x)
{
    return 4.0 / (1.0 + x * x);
};

double appPi = trapezoidal(f, 0.0, 1.0);

double tol = 1e-6;
int max_refinements = 20;
double appPi2 = trapezoidal(f, 0.0, 1.0, tol, max_refinements);

cout << "Trapezoid Rule results for computing pi by integration" << endl;
cout << "a) with defaults, and b) with tol and max_refine set : " << endl;
cout << appPi << ", " << appPi2 << endl << endl;

我提供了两个示例，一个使用默认设置来进行集成和收敛范围的离散化，另一个使用自定义设置。

我的结果（只是从屏幕上获取输出的副本）是：

Trapezoid Rule results for computing pi by integration
a) with defaults, and b) with tol and max_refine set :
3.14159, 3.14159